Consider the quadratic equation `nx^(2) +7 sqrt(nx)+n=0`, where n is a positive intergar. Which of the following statements are necessarliy correct ?
I. For any n, the roots are distinct.
(II) There are infinitely many values of n for which both roots are real.
(III) The product of the roots is necessarlity an integer.

Options

A. III only
B. I and II only
C. II and III only
D. I, II and III

Answer

b

Explanation

`D=49n-4n^(2)`
`=n(49-4n)`
`Dne0 "for any n" in I^(+)` so roots are distinct.
for roots to be real `D ge 0`
So `n le49/4`
so n can be { 1,2,3 ……….12 }
Clearly product of the roots is 1.